arX

iv:1

112.

2704

v3 [

hep-

ph]

19

Mar

201

2

A WIMPy Baryogenesis Miracle

Yanou Cui,a,b1 Lisa Randalla2 and Brian Shuvea3

a Center for the Fundamental Laws of Nature

Jefferson Physical Laboratory

Harvard University

Cambridge, MA 02138, U.S.A.b Department of Physics, University of Maryland, College Park, MD 20742, USA.

Abstract

We explore models in which weakly interacting massive particle (WIMP) dark matter annihilation isdirectly responsible for baryogenesis, thereby connecting dark matter with baryogenesis. We call thisprocess “WIMPy baryogenesis”. The dark matter relic density in these models, as with conventionalWIMP models, is obtained with only order one couplings and TeV-scale masses according to the WIMPmiracle. Thus, WIMPy baryogenesis models naturally accommodate weak-scale dark matter. Furthermore,an extension of the WIMP miracle simultaneously explains the observed baryon asymmetry and the correctdark matter abundance. The models we present have the further feature that they create the baryon numberasymmetry at the weak scale, thereby avoiding the problems in some models of baryogenesis associated withhigh reheat temperatures in supersymmetric theories. Some of these models yield observable consequencesin ongoing and future experiments.

1E-mail:cuiyo@umd.edu UMD-PP-011-0152E-mail:randall@physics.harvard.edu3E-mail:shuve@physics.harvard.edu

Contents

1 Introduction 1

2 General Analysis of WIMPy Baryogenesis 32.1 Boltzmann Equations and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Estimates of Baryon Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 WIMP Annihilation to Leptons 83.1 Model Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.1 Field Content and Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Asymmetry generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.3 Washout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.4 Boltzmann equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 WIMP Annihilation to Quarks 174.1 Model Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Experimental Constraints and Detection Prospects 205.1 Dark Matter Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.1.1 Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.1.2 Indirect Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Collider Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3 Electric Dipole Moment Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Conclusions 27

1 Introduction

Generally, baryogenesis and the establishment of the dark matter number density are treated as independentprocesses. With the notable exception of one class of models [7], it has largely been overlooked that modelswith symmetric, weakly interacting massive particle (WIMP) dark matter can connect dark matter physicswith baryogenesis. We present a new mechanism that creates such a link and is based on a simple premise: ifWIMP annihilation satisfies the Sakharov conditions, a non-zero baryon number asymmetry can be generatedfrom dark matter annihilation, and in some instances, can account for the entire observed baryon asymmetry.We call this process WIMPy baryogenesis. Our models are distinct from models of asymmetric dark matter,which propose that dark matter and baryons have their origins in a common asymmetry. In our models,the energy densities of baryons and dark matter are more loosely linked but can accommodate the observeddark-matter-to-baryon ratio.

[1, 2, 3, 4, 5, 6]We list below the Sakharov conditions and how they are satisfied in WIMPy baryogenesis:

1. Baryon number violation: WIMP annihilations violate baryon or lepton number. A preserved U(1)symmetry is allowed if the baryon asymmetry is balanced by a negative asymmetry in a decoupled sectorthat restores the net global symmetry. We have such a U(1) symmetry in most models we present.

2. CP violation: WIMP couplings to Standard Model fields violate CP .

3. Departure from thermal equilibrium: The cooling of the universe provides the necessary departure fromthermal equilibrium. Net dark matter annihilation begins around temperatures T . mDM, resulting ina small deviation of the dark matter number density from its equilibrium value. The annihilations can

1

DM

DM SM baryons

sterile

antibaryons

B

conserving

decayexotic

antibaryon

DM

DMSM baryons

B

violating

decay

exotic

antibaryon

Figure 1: Schematic diagrams showing the evolution of the asymmetry created by dark matter annihilation.(left) Model where asymmetry created in exotic antibaryons is sequestered in a sterile sector through baryon-number-conserving decays. (right) Model where asymmetry created in exotic antibaryons is converted into aStandard Model baryon asymmetry through baryon-number-violating decays.

generate a baryon asymmetry that depends on the amount of dark matter annihilation occurring duringwashout freeze-out, which is comparable to the dark matter density at that time.

We present models that satisfy all three Sakharov conditions, and that simultaneously generate the observedbaryon asymmetry and WIMP relic density. In particular, we find that there exist successful models of WIMPybaryogenesis with O(1) couplings and CP phases, and weak-scale masses for all new fields. This is an extensionof the WIMP miracle to also include baryogenesis, although we show that the size of the generated asymmetryis sensitive to the parameters in the theory and can vary by several orders of magnitude from the observedasymmetry.

Although WIMP annihilation can generate a baryon asymmetry, there are other processes that have thepotential to wash out the asymmetry, and their freeze-out is crucial to create the observed baryon asymmetry.In our models, the two leading sources of washout are inverse annihilations of baryons into dark matterand baryon-to-antibaryon processes. Washout scatterings must be suppressed to generate a sizeable baryonasymmetry because, as we show in Section 2, any asymmetry generated prior to washout freeze-out1 is rapidlydamped away. After washout processes freeze out, dark matter annihilations can efficiently create a baryonasymmetry, and the final asymmetry depends on how much dark matter remains when washout scatteringsfreeze out. Washout freeze-out must occur before that of WIMP annihilation, at which point dark matterannihilation is no longer efficient and no sizeable asymmetry can be created. Thus, we find our central result:if washout processes freeze out before WIMP freeze-out, then a large baryon asymmetry may accumulate, andits final value is proportional to the WIMP abundance at the time that washout becomes inefficient.

The early freeze-out of washout processes can occur for kinematic reasons. Inverse annihilations will beBoltzmann-suppressed for T < mDM because the thermal baryon fields are no longer energetic enough toannihilate back into dark matter. Baryon-antibaryon scatterings, however, can remain rapid at temperatureswell below mDM. The only way to suppress baryon-to-antibaryon washout is if all washout processes involve aheavy exotic baryon field in the initial state. We illustrate this scenario in Figure 1, showing how dark matterannihilates to Standard Model baryons plus an exotic baryon, as well as the possible decays of the exoticbaryon (either through baryon-preserving or baryon-violating interactions). If this exotic field has a mass& mDM, its abundance is Boltzmann-suppressed at T < mDM and suppresses the washout rate. Meanwhile,dark matter annihilations are not kinematically allowed if the heavy baryon field has mass & 2mDM, so themass condition mDM . mexotic baryon . 2mDM is essential to generate a large baryon asymmetry throughWIMPy baryogenesis.

Dark matter annihilations generate a positive baryon asymmetry stored in Standard Model quarks alongwith an equal negative asymmetry stored in the exotic baryon field. It is important that the decays of the

1The time of washout freeze-out is defined as when the rate of washout processes falls below the Hubble expansion rate. Thisis analogous to the freeze-out of WIMP annihilation.

2

exotic baryon do not eliminate the Standard Model baryon asymmetry. In models of WIMPy baryogenesis witha preserved U(1) baryon symmetry, the exotic baryon-number-carrying field is charged under an additionaldiscrete symmetry, while Standard Model fields are uncharged, preventing the exotic baryon from decayinginto Standard Model baryons and destroying the asymmetry. The heavy baryon-number-carrying field decaysinstead into light gauge singlet fields that are charged under the discrete symmetry and decoupled fromStandard Model fields at temperatures below the scale of WIMPy baryogenesis. We also present a modelwhere the exotic baryon decays to Standard Model quarks through baryon-number-violating couplings, andsuch models manifestly satisfy the first Sakharov condition.

For many years, the WIMP miracle – the fact that dark matter fields with weak-scale masses and annihila-tion cross sections give the correct dark matter thermal relic density – has been a compelling paradigm for darkmatter model-building. WIMPy baryogenesis preserves the WIMP miracle while also offering an explanationfor the observed baryon asymmetry. While WIMPy baryogenesis models do not predict the precise relation-ship between the dark matter and baryon number densities, natural models do restrict the baryon asymmetryto a range of about seven orders of magnitude (see Section 2), and the observed asymmetry is within thisrange. Since baryogenesis arises from WIMP annihilations, WIMPy baryogenesis is also necessarily connectedto weak-scale physics. While we do not discuss an embedding of WIMPy baryogenesis in a particular solutionof the hierarchy problem, we assume that whatever new physics lies at the weak scale stabilizes any scalarpotentials in our theory and gives a natural explanation for their weak-scale masses. A consequence of thisis that, with weak-scale masses, some of the new fields necessary for baryogenesis may give signals at futureexperiments. Additionally, the present-day dark matter is symmetric, leading to the possibility of the indirectdetection of dark matter annihilations as in conventional WIMP models. This is in contrast with genericasymmetric dark matter models, in which the majority of dark matter annihilations ceased long before thepresent day, although it is also noteworthy that there do exist scenarios in which the symmetric componentof dark matter is regenerated at late times, giving indirect detection signals for some asymmetric dark mattermodels [6].

A further advantage of this scenario is that bounds on the reheat temperature in supersymmetric modelsdo not constrain WIMPy baryogenesis. Typical reheat temperature constraints come from overproduction ofgravitinos and are in the range TRH . 106− 109 GeV [8], and TRH is consequently below the scale required forconventional leptogenesis through the decay of heavy, Majorana right-handed neutrinos. Although low-scalemechanisms for baryogenesis are known, such as electroweak baryogenesis, WIMPy baryogenesis is a new wayof generating the baryon asymmetry at T ∼ TeV while satisfying the reheat bound.

We discuss the general conditions for successful WIMPy baryogenesis in Section 2, finding that interactionswashing out the baryon asymmetry must become ineffective prior to WIMP freeze-out in order to generate theobserved asymmetry. In Section 3, we focus on a particular model where dark matter annihilates through alepton-number-violating interaction and the asymmetry is subsequently transferred to baryons by sphalerons.Because sphaleron processes are only rapid in the unbroken electroweak phase, such baryogenesis must occurbefore the electroweak phase transition. We compute the dark matter relic density and baryon asymmetry,and find the range of masses and couplings that agrees with the observed densities of both. We also considerthe implications of additional lepton-number-conserving dark matter annihilation channels. In Section 4, weconsider models with WIMPs annihilating directly to quarks, where baryogenesis can occur over a wider rangeof temperatures because sphalerons are no longer needed to establish the baryon asymmetry. We discussexperimental constraints and possible signals for models of WIMPy baryogenesis in Section 5. Finally, wesummarize in Section 6.

2 General Analysis of WIMPy Baryogenesis

To begin our discussion of WIMPy baryogenesis, we highlight in Section 2.1 some of its general features anduse an analytic approximation to determine the regimes in which baryogenesis is successful. Our central resultis that the final baryon asymmetry from WIMPy baryogenesis is proportional to the dark matter density atthe time when washout processes freeze out. This means that washout scatterings must freeze out at a timewhen the dark matter density was larger than or comparable to the observed final baryon asymmetry, and

3

that washout freeze-out occurs at such a time when one of the baryon-number-carrying products of WIMPannihilation is heavier than the dark matter mass, as described in the introduction. In Section 2.2, we estimatethe magnitude of the baryon asymmetry for input parameters consistent with the WIMP miracle and showthat it can lie within a range of approximately seven orders of magnitude.

2.1 Boltzmann Equations and Solutions

We consider a theory with dark matter species X whose annihilation violates baryon number, creating onequark (or anti-quark) along with other field(s) ψi. In this section we will not specify the precise interactionsmediating dark matter annihilation in order to avoid specific model dependence. X can be either Majoranaor Dirac, and the results derived below remain the same up to O(1) multiplicative factors. All manifestationsof the Boltzmann equation for baryon number evolution in WIMPy baryogenesis models have two importantterms: one that describes the annihilation of dark matter and the consequent generation of a baryon asymmetry,and another that drives the asymmetry towards its equilibrium value of zero through baryon-number-violatingwashout scatterings. The full Boltzmann equations describing the evolution of the various particle abundancesare model-dependent and can have many terms, which we give explicitly for concrete models in Sections 3 and4. However, in the models of interest to us, namely models where the asymmetry arises predominantly fromWIMP annihilations, the overall dynamics are well-described by the inclusion of only these terms.

Consider the limit where WIMP annihilations are the dominant source of the baryon asymmetry and forwhich the asymmetry is small as observed. We derive the Boltzmann equations in terms of dimensionlessquantities: the number density per comoving volume of field i, Yi = ni/s (s is the entropy density), and thetemperature, which we express as x = mX/T . The dark matter number density is denoted YX and the baryonasymmetry is denoted Y∆B. The YX evolution equation has one term that is important in all models of WIMPybaryogenesis, namely the conventional WIMP annihilation term that is proportional to the annihilation crosssection σann and drives YX to its equilibrium value. This term arises from both XX → baryon processes andthe inverse processes, baryons → XX . The YX Boltzmann equation with this term is

dYXdx

= − 2s(x)

xH(x)〈σannv〉

[

Y 2X − (Y eq

X )2]

, (1)

where H(x) is the Hubble scale.We neglect a back-reaction term in the YX Boltzmann equation ǫ s(x) 〈σannv〉Y∆B(Y eq

X )2/(2Yγ xH(x)),where ǫ is the net baryon number created per dark matter annihilation and is a measure of the magnitude ofCP -violation (it is defined more precisely in Section 3). This term accounts for the modification of the inversescattering rate of baryons into X when there is a baryon asymmetry. This approximation is valid because thisterm is small when Y∆B ≪ 1, as is true in our universe (Y∆B ∼ 10−10). This simplification also decouplesthe equations for YX and Y∆B, which makes it easier to get an approximate analytic solution for Y∆B. Theequation (1) in this limit is the same as the familiar Boltzmann equation for conventional WIMPs. YX is thenobtained from the standard WIMP relic density calculation [9] and is approximately inversely proportional tothe annihilation cross section.

The Boltzmann equation for the evolution of the baryon asymmetry has two important terms. In thefirst term, a baryon asymmetry is generated through X annihilations, and is proportional to ǫ/2 × dYX/dx,which is the annihilation rate multiplied by the fractional asymmetry generated per annihilation. The factor of1/2 arises because the annihilation term in (1) includes the sum of annihilation into baryons and antibaryons,whereas the term generating the asymmetry includes the difference. The second term in the baryon asymmetryBoltzmann equation reduces the existing baryon asymmetry and is the washout term. It is proportional to Y∆Bmultiplied by the cross section of processes that eliminate the baryon asymmetry σwashout. The Boltzmannequation is

dY∆Bdx

=ǫ s(x)

xH(x)〈σannv〉

[

Y 2X − (Y eq

X )2]

− s(x)

xH(x)〈σwashoutv〉

Y∆B2Yγ

∏

i

Y eqi . (2)

The factor of Y∆B/2Yγ comes from the the fact that the chemical potential µ∆B for the baryon asymmetrycan be written as µ∆B/T = Y∆B/2Yγ [9]. We assume that all species except for X are in equilibrium. There

4

are other terms that we have not included, such as washout terms proportional to Y eqexotic−B YX that come from

scattering of baryon-number-carrying fields off dark matter fields. Typically, the suppression coming from thesmall value of YX for x ≫ 1 makes this term subdominant to other washout terms. In the models of Section3 and 4, the Y eq

exotic−B YX term can be ignored without substantially affecting the numerical results.As expected, the Boltzmann equations show that the total baryon number is zero when all fields are in

equilibrium and with the initial condition Y∆B = 0. A solution for Y∆B can be written in integral form interms of the X density:

Y∆B(x) =

∫ x

0

dx′ǫ s(x′)

x′H(x′)〈σannv〉

[

Y 2X − (Y eq

X )2]

(x′) exp

[

−∫ x

x′

dx′′

x′′s(x′′)

2Yγ H(x′′)〈σwashoutv〉

∏

i

Y eqi (x′′)

]

(3)

≈ − ǫ

2

∫ x

0

dx′dYX(x′)

dx′exp

[

−∫ x

x′

dx′′

x′′s(x′′)

2Yγ H(x′′)〈σwashoutv〉

∏

i

Y eqi (x′′)

]

. (4)

Equation (4) explicitly shows that Y∆B(x) can be expressed in terms of a source term from dark matter anni-hilations, and an exponential term that attempts to erase any asymmetry generated by WIMP annihilations.The source term can be written as dYX/dx, as in [10]. At T & mX , or x . 1, WIMP annihilations are balancedby inverse scattering processes and dYX/dx ≈ 0, meaning no asymmetry is generated according to (4). Atx & 1, the expansion and cooling of the universe result in net WIMP annihilations (dYX/dx 6= 0), providingthe departure from equilibrium necessary for baryogenesis. The net asymmetry at any x is sensitive to therate of washout processes during the epoch of WIMP annihilations.

The integrand in the exponent (4) is the washout rate Γwashout(x) normalized to the Hubble scale H(x),

Γwashout(x)

H(x)=

s(x)

2Yγ H(x)〈σwashoutv〉

∏

i

Y eqi (x). (5)

Washout freezes out when Γwashout/H < 1. In the limit where Γwashout is a rapidly decreasing function of x,(4) takes a particularly simple form. This is true if, for example, the washout rate freezes out because mi/Tbecomes large and yields an exponential suppression of Y eq

i . In this case, we can model the exponential in (4)as a step function and obtain

Y∆B(∞) ≈ − ǫ

2

∫ ∞

xwashout

dx′dYX(x′)

dx=ǫ

2[YX(xwashout)− YX(∞)] , (6)

where xwashout = mX/Twashout is the point at which washout processes freeze out, and YX(∞) is the late-timedark matter relic density.

Equation (6) has a very clear physical interpretation: after washout scatterings freeze out, all subsequentWIMP annihilations generate a baryon asymmetry with efficiency ǫ. This is why, according to (6), Y∆Bis proportional to ǫ times the total number of WIMP annihilations that happen after xwashout, which isYX(xwashout) − YX(∞). The observed baryon asymmetry is Y∆B ≈ 9× 10−11 [11]. Since dark matter at latetimes satisfies the relation

YX(∞) ≈ (5 GeV)Y∆B(∞)

mX

, (7)

we require that YX(∞) < Y∆B(∞) for weak-scale dark matter. Along with the requirement ǫ < 1, (7) and (6)imply that YX(xwashout) ≫ YX(∞). In other words, the washout interactions must become ineffective priorto XX annihilation freeze-out in order to generate a sufficiently large baryon asymmetry through WIMPybaryogenesis. As an example of the numerical scales in WIMPy baryogenesis: for a WIMP of mass 1 TeV,YX(∞) ≈ 4 × 10−13 and WIMP freeze-out happens at xf.o. ≈ 27, or T ≈ 37 GeV. For ǫ = 0.1, washoutscatterings must freeze out at xwashout ≈ 20 or T ≈ 50 GeV. The final baryon asymmetry is proportional tothe WIMP density at the time when washout ceases to be important, with Y∆B(∞) ≈ 9× 10−11.

For what parameters do we expect washout processes to freeze out prior to WIMP annihilation freeze-out?We compare Γwashout in (5) to the corresponding rate of WIMP annihilation, which is [9]

ΓWIMP(x)

H(x)=

2s(x)

H(x)〈σann v〉YX(x). (8)

5

We then find thatΓwashout(x)

ΓWIMP(x)≈ 〈σwashoutv〉

∏

i Yeqi (x)

4〈σann v〉Y eqX (x)Yγ

. (9)

This ratio must be less than one at the time of washout freeze-out for washout processes to freeze out priorto WIMP freeze-out. This can be realized if either of the following is true for every process washing out thebaryon asymmetry:

1. One of the baryon states is heavier than dark matter so∏

i Yeq

i(x)

Yeq

X(x)Yγ

≪ 1.

2. The baryon-number-violating coupling is small so 〈σwashout v〉 ≪ 〈σann v〉.The second scenario is challenging to realize, because the same baryon-number-violating couplings appearin both the washout and annihilation cross sections, and 〈σann v〉 is fixed by the dark matter relic density.Furthermore, as we show in Section 3.1.3, suppressing the washout cross section also suppresses the fractionalasymmetry generated per annihilation, ǫ, and the resulting baryon asymmetry is typically too small. Therefore,we expect that viable models of WIMPy baryogenesis have at least one baryon-number-carrying field with mass& mX .

2.2 Estimates of Baryon Asymmetry

In this section, we derive an estimate of the baryon asymmetry generated by a WIMP dark matter candidatewith mass mX ∼ TeV, and we determine the size of the baryon energy density compared to the WIMP relicdensity. In the following, we assume for simplicity that a dark matter field X annihilates into a StandardModel quark Q plus an exotic baryon field ψ (see Sections 3 and 4 for specific model details). We find thatthe baryon asymmetry depends strongly on the mass mψ and is constrained to lie within a seven or eightorder-of-magnitude window, with the observed baryon asymmetry within an order of magnitude of the upperlimit. Therefore, WIMPy baryogenesis does not predict the value of the dark matter-baryon ratio, but neitheris the relationship between the two energy densities completely arbitrary.

To determine the range of baryon asymmetries obtained from WIMPy baryogenesis, we use the result fromthe last section that the final baryon asymmetry is proportional to the number of dark matter annihilationsthat occur after washout freeze-out, as shown in (6). The largest possible asymmetry is generated when theexotic baryon field is heavy relative to dark matter (mψ & mX) so that washout processes freeze out whilethere is still a large dark matter abundance. To determine the upper bound on the asymmetry, we use the factthat mψ < 2mX for WIMP annihilation to be allowed kinematically, and this limits how many dark matterparticles can remain when washout freezes out. By contrast, the baryon asymmetry is small when washoutprocesses turn off at a late time (mψ ≪ mX) after dark matter annihilation has frozen out. To calculate thelower bound, we determine the rate of residual dark matter annihilation after dark matter freezes out and usethis to determine the size of the asymmetry.

For both the upper and lower limits, we first calculate the allowed baryon asymmetry and then determinethe corresponding dark matter-baryon ratio. In both scenarios, the baryon asymmetry depends on two timescales: the point of washout freeze-out, xwashout = mX/Twashout, and the point at which WIMP annihilationfreezes out, xann = mX/Tann.

Estimate of upper limit: We first estimate the upper limit of the baryon asymmetry generated withinour framework, which occurs when mψ is heavy to suppress washout and is therefore also at the TeV-scale.Kinematically, dark matter annihilation occurs only if mψ < 2mX , which bounds how early xwashout can berelative to xann. For a TeV-scale dark matter field, WIMP annihilation freezes out when the temperature isabout 1/30 of its mass. Therefore xwashout ≈ xann(mX/mψ) & 15. We also know that, when washout freezesout while WIMP annihilation is still active, YX(xwashout) ≫ YX(∞). We then obtain from (6):

Y∆B(∞) ≈ ǫ

2YX(xwashout) <

ǫ

2Y eqX (15) ≈ ǫ× 10−8. (10)

According to (10), the asymmetry is independent of mX and depends only on the ratio xwashout ≈ mψ/mX ,with a large asymmetry when mψ is comparable to or larger than mX .

6

To compare the baryon density to the dark matter energy density, recall that the WIMP density changeslittle after annihilation freezes out, and so YX(∞) ≈ YX(xann) ≈ YX(30). We then find that

ΩBΩX

=mproton Y∆B(∞)

mX YX(∞)≈ ǫ

2

YX(xwashout)

YX(xann)

(

GeV

mX

)

≈ ǫ

2

Y eqX (15)

Y eqX (30)

(

GeV

mX

)

. 10( ǫ

10−2

)

(

TeV

mX

)

. (11)

Therefore, for a model with weak-scale mX , O(1) couplings (in accordance with the WIMP miracle), and theloop-suppressed ǫ ∼ 10−2 as in (24), we find that the energy density of baryons can be at most an order ofmagnitude larger than the energy density of dark matter.

Estimate of lower limit: In deriving the upper bound, we assumed thatmψ saturated the bound mψ < 2mX

and we found that dark matter annihilation could generate the observed baryon asymmetry. Whenmψ ≪ 2mX ,washout processes remain in equilibrium until after dark matter freeze-out, and the asymmetry from WIMPybaryogenesis is too small to account for the observed asymmetry. We now estimate the full range of baryonasymmetries achieved in our models when mψ ≪ 2mX . In this case, the equilibrium number density of X ismuch smaller than the actual, frozen-out X abundance. As a result of this overabundance of X relative to itsequilibrium value, some residual dark matter annihilations continue at late times, even though the annihilationrate is insufficient to appreciably change YX after xann. Such annihilations can, however, generate a smallbaryon asymmetry. According to (6), this asymmetry can be estimated by calculating YX(∞)− YX(xwashout),where xwashout > xann.

To determine the asymmetry, we solve the Boltzmann equation (1), neglecting the subdominant term(Y eqX )2 in equation (1). Furthermore, if XX annihilation is s-wave, then 〈σann v〉 is approximately constant in

the domain xann < x < xwashout. The only x-dependence comes from the factor

s(x)

xH(x)=

s(xann)

H(xann)

xannx2

. (12)

Integrating (1) from x = xwashout to x = ∞ gives

YX(xwashout)− YX(∞) ≈ 2s(xann)xann 〈σann v〉YX(xann)2

H(xann)

1

xwashout. (13)

Using the definition of ΓWIMP in (8), together with the fact that ΓWIMP(xann) = H(xann), gives the simpleresult

YX(xwashout)− YX(∞) ≈ 2xannxwashout

YX(xann) ≈2xannxwashout

YX(∞). (14)

Notice that for xwashout > xann, YX is constant at leading order from xwashout to ∞. Also as mentioned earlier,by assuming both X and ψ have weak-scale masses and interactions, xann/xwashout ∼ mψ/mX . We can thenobtain an estimate for the baryon asymmetry:

Y∆B ≈ ǫ xannxwashout

YX(∞) ≈ ǫ

(

mψ

mX

)

YX(∞). (15)

We see that the baryon asymmetry decreases linearly with mψ when mψ ≪ mX .The ratio of the baryon energy density to the dark matter energy density is

ΩBΩX

∼ 10−3 × ǫ

(

mψ

mX

)(

TeV

mX

)

. (16)

If there is no large hierarchy in mX and mψ (i.e. mψ/mX & 0.1), and using our earlier estimate of ǫ ∼ 10−2

for O(1) couplings that give the correct WIMP relic density, we find that ΩB/ΩX & 10−6. We emphasize,however, that even smaller asymmetries are possible if the imaginary parts of the couplings are tuned to besmall or if there exist hierarchies in the masses of the new fields.

7

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!"#$%*.3,45$&(*+,-./$

!2'6

Figure 2: The evolution of the number density per comoving volume for field i (Yi) as a function of x = mX/T .The numerical solutions shown here are based on the WIMPy leptogenesis model discussed in Section 3, wherethe dominant annihilation process is XX → Lψ and the dominant washout is Lψ → L†ψ†. The inputparameters are yX = 2.7, λL = 0.8, ǫ = 0.2, mX = 3 TeV, and mS = 5 TeV. mψ = 4 TeV gives the behaviorwhen washout freezes out well before WIMP annihilation freezes out (“weak washout”). mψ = 2 TeV givesthe behavior when washout becomes ineffective subsequent to WIMP freeze-out (“strong washout”).

Considering equations (11) and (16), we find that the expected range for the baryon-to-dark matter ratioin WIMPy baryogenesis is

10−6 .ΩBΩX

. 10, (17)

assuming O(1) couplings, and weak-scale masses for all new fields, i.e. mX ,mψ ∼ O(0.1 − 1TeV). The ob-served value of ΩB/ΩX ≈ 0.2 falls within this range, and thus WIMPy baryogenesis can account for the entireobserved baryon asymmetry, but it does fall toward the upper end of the allowed region.

To summarize, models of WIMPy baryogenesis predict a dark matter relic density inversely proportional tothe WIMP annihilation cross section, as in conventional WIMP models, and a baryon asymmetry proportionalto the dark matter density at the time when washout processes freeze out. In Figure 2, we illustrate theevolution of the dark matter abundance and the baryon asymmetry in one model of WIMPy baryogenesis forthe two limiting washout cases.

3 WIMP Annihilation to Leptons

3.1 Model Overview

We have discussed baryogenesis in the generalized sense of either the direct production of a baryon asymmetrythrough WIMP annihilation or leptogenesis, in which a lepton asymmetry is produced by WIMP annihilationand converted to a baryon asymmetry through sphalerons. In this section, we present a model of leptogenesis,where the lepton asymmetry is generated above the electroweak phase transition while sphalerons are stillactive. We discuss the field content and symmetries of the model Section 3.1.1, and we calculate the efficiencyof generating a lepton number asymmetry in Section 3.1.2. As we showed in Section 2, the final baryonasymmetry is determined by the time at which washout processes freeze out. We address washout in Section3.1.3, discussing the implications for the WIMPy leptogenesis parameter space. Finally, we give the Boltzmannequations in Section 3.1.4.

8

X X ψ ψ S n Standard Model

Z4 +i −i −1 −1 −1 −1 +1

Table 1: Z4 charges of fields in the WIMPy leptogenesis model described by (18).

3.1.1 Field Content and Lagrangian

We consider a simple model with the minimal ingredients for WIMPy leptogenesis. Dark matter consists ofa pair of gauge singlet Dirac fermions2 X and X that annihilate to the Standard Model lepton doublet Liand new weak-scale fields ψi. X annihilates through weak-scale gauge singlet pseudoscalars3 Sα. By gaugeinvariance, ψi has charge (2, 1/2) under the SU(2)L ×U(1)Y gauge interactions. The Lagrangian is

L = Lkin + Lmass −i

2

(

λXαX2 + λ′XαX

2)

Sα + i λLαi SαLiψi + h.c. (18)

To satisfy the Sakharov conditions, dark matter annihilation must also violate CP , and λLαi must be complex.To have physical CP violation, there must be more than one scalar Sα so there is a relative phase in theiramplitudes. XX annihilation can then generate an asymmetry in Li, and the lepton asymmetry is subsequentlyconverted to a baryon asymmetry by sphalerons. Because the symmetry preserved by sphalerons is B − L, anegative lepton asymmetry must be generated to account for the observed positive baryon asymmetry.

A positive lepton number asymmetry also accumulates in ψi, and it is important that this positive asymme-try does not erase the negative asymmetry in Standard Model leptons. In our model, ψi decays into light gaugesinglets ni that are decoupled from Standard Model fields at low temperatures. The asymmetry produced inψ is therefore sequestered in a sterile sector and the Standard Model asymmetry persists to the present time4.A Z4 symmetry, with charges in Table 1, forbids other operators that allow ψ to decay directly into StandardModel leptons, thus preventing the erasure of the Standard Model lepton asymmetry. The Z4 symmetry alsomakes dark matter stable.

In the simplest model, ψi decays to ni +H through the interaction

∆L = λiH†niψi + h.c. (19)

We assume that ψi is vectorlike with a partner ψi in order to more readily satisfy electroweak precisionconstraints. Its mass is restricted by the LEP bound mψ & 100 GeV (see Section 5.2).

After electroweak symmetry breaking, ψi mixes with the sterile neutrino ni, and we must ensure thatthe sterile neutrino satisfies overclosure constraints. Since ψ is Dirac, we also include ψ when diagonalizingthe mass matrix and find that there remains a massless eigenstate even after the Higgs condenses. This isgood, because light, weakly interacting thermal relics (such as sterile neutrinos) with masses & O(eV) wouldoverclose the universe. n could have a Majorana mass mn . eV and still satisfy observational constraints,since the light eigenstate would have a mass . eV as well, but we take n to be massless in our model.

The Lagrangian (18) is also invariant under a U(1)3 lepton flavor symmetry that prohibits flavor-changingneutral currents but allows flavor-dependent couplings. Li, ni, and ψi have charges +1, +1, and −1, respec-tively, under the U(1)i factor of the flavor symmetry. We assume that the only source of flavor-breaking inthe low-energy theory is through the neutrino mass matrices, and this effect is very small.

2A Majorana dark matter field X does not work in this case because X must carry a complex charge for the model to generatea non-zero lepton asymmetry, as we show later in this section.

3We consider pseudoscalars instead of scalars because they do not have a velocity-suppressed XX annihilation cross section.A scalar S with couplings to X that are CP -violating with a large imaginary part would work as well.

4If mψ > mS , then ψ can decay into S + L† and wipe out the lepton asymmetry. However, S then subsequently decays into

either L + H + n† or L† + H∗ + n (when XX → Lψ is kinematically allowed, S → XX is kinematically forbidden because2mX > mψ > mS), and the difference in the rates of these decays generates another lepton asymmetry. If the efficiency ofasymmetry generation from S decays is comparable to that from XX annihilations, the asymmetry is comparable to the originalasymmetry created from XX → Lψ.

9

ψ

L

S1

ψ

L

S1S2

L†

ψ†ψ

L

S1 S2

ψ†

L†

Figure 3: Diagrams of tree and loop contributions to S decay. The difference between these rates and theirconjugates generates a lepton asymmetry.

X

X ψ

L

S1

X

X ψ

L

S1 S2

L†

ψ†

X

X ψ

L

S1S2

ψ†

L†

Figure 4: Diagrams of tree and loop contributions to the XX annihilation cross section. The difference betweenthese rates and their conjugates generates a lepton asymmetry.

3.1.2 Asymmetry generation

In this model, a lepton asymmetry can in principle be generated through two processes: the more conventionalprocess of Sα decay into Liψi and their conjugates (or directly into Li + ni + H if mψ > mS), and XXannihilation into the same final states. We show these in Figures 3 and 4, respectively, assuming that decayand annihilation occur predominantly through the lightest scalar S1. Existing work discusses the relevantprocesses for generating a lepton asymmetry through 2 → 2 scattering [12], although the authors consideronly high-scale models (T & 109 GeV) with qualitatively different features than WIMPy leptogenesis. CP -violating phases in our model appear in the interference between tree-level and one-loop diagrams. We defineasymmetry factors for the decay of the lightest scalar S1 and for WIMP annihilations, respectively, in themanner of conventional leptogenesis:

ǫ1 =Γ(S1 → ψiLi)− Γ(S1 → ψ†

iL†i )

Γ(S1 → ψiLi) + Γ(S1 → ψ†iL

†i ), (20)

ǫ2 =σ(XX → ψiLi) + σ(XX → ψiLi)− σ(XX → ψ†

iL†i )− σ(XX → ψ†

iL†i )

σ(XX → ψiLi) + σ(XX → ψiLi) + σ(XX → ψ†iL

†i ) + σ(XX → ψ†

iL†i ). (21)

ǫ1 gives the fractional asymmetry generated per S1 decay, while ǫ2 gives the fractional asymmetry generatedper XX annihilation. The precise values of ǫ1, ǫ2 in this case depend on the masses mSα and the couplingsλαi.

To reduce the number of arbitrary parameters in our analysis, we make the following assumptions:

• Dark matter annihilation occurs dominantly to only one flavor of lepton, and the couplings of all otherleptons to Sα are zero. The non-zero couplings of the single lepton flavor are denoted λLα.

• Dark matter annihilation and washout occur mostly through the lightest scalar, S1, and we consider therates of only these processes in our analysis. For concreteness, we require that the corresponding crosssections with intermediate S2 to be less than 20% of the corresponding cross sections with S1, giving

10

roughly

λ4L2m4S2

.λ4L15m4

S1

, (22)

λ2X2λ2L2

m4S2

.λ2X1 λ

2L1

5m4S1

. (23)

We also assume that mS1 ≪ mS2, so that the loop integrals in ǫ1 and ǫ2 can be put in a simple analyticform (24).

• The physical CP phases are large i.e. Im(a) ≈ a where a is some product of couplings appearing inscattering and decay amplitudes.

None of these assumptions are required byWIMPy leptogenesis, and we make them only to simplify the analysisand its interpretation. Relaxing these assumptions would introduce much complexity into the Boltzmannequations while giving qualitatively similar results. The phenomenology does, however, depend to some extenton the flavor of leptons to which dark matter predominantly annihilates (see Section 5 for details). With theabove assumptions5,

ǫ2 ≈ − 1

6π

Im(λ2L1λ∗2L2)

|λL1|2(2mX)

2

m2S2

[

7− 15

(

mψ

2mX

)2

+ 9

(

mψ

2mX

)4

−(

mψ

2mX

)6]

. (24)

The expression for ǫ1 is the same but with 2mX → mS1. Since we are most interested in the asymmetryfrom annihilation, ǫ2 is the relevant parameter for WIMPy baryogenesis and we denote its asymmetry factorby ǫ ≡ ǫ2. ǫ is suppressed by 1/m2

S2 from the S2 propagator, and is proportional to (2mX)2 because the

momentum flowing through the S1 propagator in XX annihilation is√s = 2mX+O(T ), where T ≪ mX ,mS1

at freeze-out. Note that (24) vanishes when mψ = 2mX , at which point the particles in the loop cannot go onshell and there is no imaginary part of the amplitude (and, hence, no CP violation).

Using (22), (24), and the assumption of large CP phases, we can bound ǫ from above:

|ǫ| . 2λ2L13π

√5

m2X

m2S1

[

7− 15

(

mψ

2mX

)2

+ 9

(

mψ

2mX

)4

−(

mψ

2mX

)6]

. (25)

We treat ǫ as a free parameter, subject to (25), and we can now express all rates and cross sections in termsof λX ≡ λX1, λL ≡ λL1, ǫ, mX , mψ, and mS ≡ mS1.

We have assumed that the lepton asymmetry from XX annihilations dominates over that from S decays.We find that this assumption is true whenever mX < mS . Since the asymmetry is proportional to the numberdensity of X or S at the time of washout freeze-out, the ratio of asymmetry from decay vs. annihilation is thesame as the ratio of the number of S particles to the number of X particles at the time of washout freeze-out.The assumption of annihilation-dominated asymmetry is therefore equivalent to mX < mS .

3.1.3 Washout

As we demonstrated in Section 2, the final baryon asymmetry depends on the time of washout freeze-out.We now discuss the implications for WIMPy leptogenesis, finding that we need mψ & mX for successfulWIMPy leptogenesis. We show the lepton number washout processes in Figure 5. They include inverseannihilations, lepton → antilepton scatterings, and ψX → L†X processes. The dominant washout is typically

5This expression is derived in the narrow-width approximation. For TeV WIMPs, λX & 1 is often necessary to obtain the correctdark matter relic abundance, which may lead to ΓS1, ΓS2 ∼ mS . When 2mX > mS1, S1 is kinematically forbidden from decayinginto XX and the S1 width is narrow (because typically λL . 1). S2 may be broad, but the imaginary part of its self-energycorrection ImΠ(p2) as substituted into (24) must be evaluated at p2 = 4m2

X≪ m2

S2 and satisfies ImΠ(4m2X) ∼ 4m2

X≪ m2

S2.Similarly, if 2mX < mS1, then the partial width of S1 to X can be very large for 2mX ≪ mS1, but once again ImΠ is evaluated in(24) as ImΠ(4m2

X) ∼ 4m2

X< m2

S1. Therefore, the narrow-width approximation holds true to a degree sufficient for our purposes.

11

Figure 5: Diagrams leading to washout of the lepton number from (top row) s-channel and (bottom row)t-channel scatterings.

from Lψ → L†ψ† scatterings, because the inverse annihilation Lψ → XX is kinematically suppressed forT < mX and ψX → L†X gets more Boltzmann suppression. Applying (5) for the specific model of WIMPyleptogenesis, the washout rate is proportional to

Γwashout(x) ≈s(x)

Yγ〈σLψ→L†ψ† v〉Y eq

L Y eqψ (x), (26)

where s(x) is the entropy density at x. Washout freezes out when its rate is about equal to the Hubble scale,Γwashout(xwashout) ≈ H(xwashout). Γwashout(xwashout) can be small for one of two reasons:

1. mψ & mX so that Y eqψ (xwashout) is Boltzmann-suppressed while dark matter is annihilating.

2. 〈σLψ→L†ψ† v〉 is small relative to the annihilation cross section so that washout freezes out before anni-hilation. The washout cross section can be small if λL ≪ 1.

One of these two conditions must hold for each washout process. We find that option #1 leads to viableWIMPy leptogenesis. Option #2, on the other hand, does not give a large asymmetry. According to (25),the asymmetry efficiency factor ǫ is also suppressed when λL ≪ 1, and the potential gain in the baryonasymmetry from early washout freeze-out in option #2 is offset because leptogenesis occurs less efficiently.Therefore, mψ & mX is generally required to generate the observed baryon asymmetry.

Whenmψ is much larger thanmX (we find this is typically true formψ & 2mX), the exponential suppressionof Yψ is so large that 3 → 3 scatterings of LnH → L† n†H∗ become important (see Figure 6). This regionis, however, kinematically inaccessible in WIMPy leptogenesis since 2mX > mψ for efficient annihilation tooccur, and we neglect 3 → 3 processes.

3.1.4 Boltzmann equations

We consider the evolution of a single component La, where a is a gauge index (flavor indices are suppressedsince we consider only one flavor). We define BrL (BrX) as the total branching fraction of S into leptons(X), with BrL + BrX = 1. Also, ξ = 1 + µψa

/µLa, where µ are chemical potentials, and η is defined as

12

n†

H

L L†

H∗

n

S

ψ†ψ

Figure 6: 3 → 3 washout process that can dominate over 2 → 2 scattering when T ≪ mψ.

the amount of La asymmetry generated for each La directly created/annihilated (this accounts for the factthat the asymmetry spreads among all baryons and leptons by rapid thermalization). We calculated the crosssections and widths analytically and checked them with CalcHEP [13].

The Boltzmann equations describing the evolution of the various particle species and the asymmetry inone of the components of the La doublet are

H(mX)

x

dYX

dx= −4s〈σXX→Laψa v〉[Y 2

X − (Y eqX )2]− 2sǫ

ξ Y∆La

Yγ〈σXX→Laψa v〉(Y eq

X )2

−Br2X〈ΓS〉YeqS

(

YX

YeqX

)2

+ BrX〈ΓS〉 (YS − BrL YeqS )− ǫ

ξ Y∆La

2YγBrXBrL〈ΓS〉Y

eqS ; (27)

H(mX)

x

dYS

dx= −〈ΓS〉YS + 〈ΓS〉Y

eqS

[

BrL +BrX

(

YX

YeqX

)2]

; (28)

H(mX)

x η

dY∆La

dx=

ǫ

2BrL〈ΓS〉

[

YS + YeqS

(

1− 2BrL − BrX

[

1 +Y 2X

(Y eqX )2

])]

+ 2s ǫ〈σXX↔Laψa v〉[

Y2X − (Y eq

X )2]

−ξ Y∆La

Yγ

[

s 〈σXX↔Laψa v〉(Y eqX )2 + 2s[〈σ

Laψa↔L†aψ

†av〉+ 〈σ

(a 6=b)

Laψa↔L†bψ

†b

v〉]Y eqL Y

eqψ + 2s 〈σ

Laψb↔L†bψ

†av〉Y eq

L Yeqψ

]

−ξ Y∆La

Yγ

[

s 〈σXψa↔XL

†av〉YXY

eqψ + 2s 〈σ

ψaψa↔L†aL

†av〉(Y eq

ψ )2 + 2s 〈σ(a 6=b)

ψaψb↔L†aL

†b

v〉(Y eqψ )2

]

+ξ Y∆La

4YγBrL〈ΓS〉Y

eqS

(

ǫ2BrL + BrX

)

. (29)

We assume that all abundances are in thermal equilibrium at x = 1 and that all fields remain in equilibriumexcept for S and X . In the evolution of the scalar S, we only include the decay terms, as they dominate overSS annihilation for T ≪ mS .

To determine the relationship between µψaand µLa

, we assume that all abundances other than S, X and∆La are in thermal equilibrium and that all processes except those involving S are in chemical equilibrium.We also take sphaleron processes to be in equilibrium. The non-S couplings in (18) distribute the L and ψasymmetries among the light fields. Solving the chemical potential relations gives

ξ =16 + 12neq

ψa/neq

La

3 + 12neqψa/neq

La

, (30)

η =2(7 + 28neq

ψa/neq

La)

79 + 355neqψa/neq

La

. (31)

The precise values of ξ and η are model-dependent.

13

The assumption of thermal equilibrium for ψ is consistent provided the decay width Γψ > H(Tlep), whichin our model constrains

λ2i &16πH(Tlep)

mψ i

≈80

√g∗ T

2lep

MPlmψ i

. (32)

For Tlep = 100 GeV and mψ i = 2 TeV, this gives λi & 3 × 10−8. This is not a very stringent requirement,since this value is smaller than any of the Standard Model Yukawa couplings.

The final lepton asymmetry is also determined by the chemical potential relations. The relation betweenthe total lepton asymmetry ∆Ltot and the asymmetry in a single component of the doublet field ∆La asdetermined by equation (29) is

Y∆L tot =51 + 243neq

ψa/neq

La

7 + 28neqψa/neq

La

Y∆La. (33)

The final baryon asymmetry Y∆B follows from the sphaleron chemical potential relations, and is

Y∆B(x) = −4

(

7 + 28neqψa/neq

La

51 + 243neqψa/neq

La

)

Y∆L tot = −4Y∆La. (34)

In the limit x→ ∞, the ratio of total baryon to lepton number reduces to the same expression as conventionalleptogenesis [14].

The total dark matter relic abundance is

YDM(∞) = YX(∞) + YX(∞) = 2YX(∞). (35)

3.2 Numerical Results

There are six free parameters in our model: three masses (mS , mX , and mψ) and three dimensionless param-eters (λX , λL, and ǫ). To determine over what range of parameters WIMPy leptogenesis can be successful, weperform scans over two parameters at a time while holding others fixed. In particular, we are interested to seewhat range of masses is allowed, and if any tuning of the mass and coupling constant relations is necessary togenerate the correct baryon asymmetry and WIMP relic density.

Range of allowed masses: We hold mS fixed and determine for which mX and mψ masses there exists someperturbative couplings that give the observed dark matter density and baryon asymmetry. We place no otherrestrictions on the couplings. If mψ > mS , we assume that the S width is dominated by the three-bodydecay S → LH n. We show in Figure 7 the masses that give rise to successful WIMPy leptogenesis. Theviable ψ masses satisfy mψ ≈ (1 − 2)mX , while there is no correlation between mX and mS as long asmX < mS . For smaller values of mψ/mX , the Boltzmann suppression of the washout rate is insufficient togenerate the observed baryon asymmetry, while mψ & 2mX is not allowed because dark matter annihilationis not kinematically allowed and because the asymmetry efficiency ǫ is zero (CP violation is zero if L and ψcannot go on-shell in the dark matter annihilation loop diagrams).

The lower boundary of the allowed region has a meandering shape around mψ ≈ mS . The reason is that s-channel washout processes have a resonant enhancement in this region, leading to a smaller baryon asymmetryand a restricted parameter space. Above resonance, t-channel washout processes are also important, explainingwhy the bend in the curve is centered at mψ slightly larger than mS .

The principal reason that it is difficult to generate a large baryon asymmetry is because the efficiency ofasymmetry generation ǫ is tied to the washout cross section through its dependence on λ2L in (25). A largeasymmetry can only be generated when washout effects are also large, limiting how much of an asymmetrycan be generated. The viable parameter space is larger if (25) can be relaxed, as is the case when S1 and S2

are nearly degenerate and the asymmetry is resonantly enhanced, but this is not a required feature of WIMPyleptogenesis.

In leptogenesis, the asymmetry must be generated prior to the electroweak phase transition, at which pointsphalerons decouple and the conversion of a lepton asymmetry into a baryon asymmetry ceases. Since WIMPy

14

2mX < mΨ

Viable

paramete

rs

Washout too strong

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

mX mS

mΨ

mS

mS = 5 TeV

Figure 7: Regions in the mX -mψ plane with the correct WIMP relic density and baryon asymmetry fromWIMPy leptogenesis, with mS = 5 TeV and some choice of perturbative couplings. The masses giving bothobserved abundances are shown in blue (middle stripe). We plot the ratios mX/mS , mψ/mS to show therelationship between the X and ψ masses and the mediator scale mS . The excluded regions are shown in red:the upper region is not viable because 2mX < mψ and the thermal annihilation cross section is Boltzmann-suppressed, while the lower region has Yψ too large to prevent rapid washout of the asymmetry. The dashedline indicates the lower boundary of allowed mX and mψ; below the line, the electroweak phase transitionoccurs before the baryon asymmetry is large enough to account for the observed value. For mX/mS > 1, theasymmetry is dominated by S decay.

leptogenesis is a weak-scale model, the timing of the asymmetry generation relative to the phase transitionis important. To illustrate this, we computed the critical temperature Tc of the phase transition assuming aStandard Model Higgs with mass mh = 120 GeV, and we required that the baryon asymmetry at Tc be equalto the observed asymmetry6. This typically yields a much smaller baryon asymmetry than lepton asymmetryat late times because the baryon asymmetry stops accumulating at Tc. Accounting for the effects of the phasetransition, the allowed region is above the dashed line in Figure 7. If the phase transition is modified byadditional Higgs fields or other new physics, then this boundary line changes.

Range of allowed couplings: We choose representative values of the masses, with mS = 5 TeV for all cases,and mX and mψ chosen in the middle of the allowed bands in Figure 7. For one set of parameters, dark matterannihilates above the S resonance, with parameters mX = 4.25 TeV, mψ = 7.5 TeV, and |ǫ| = 0.075, and wedetermine the dark matter relic abundance and baryon asymmetry as functions of the two couplings. We alsostudy XX annihilation below resonance, with mX = 1.5 TeV, mψ = 2.25 TeV, and |ǫ| = 0.0075. We plot theresults in Figure 8 as contours of constant relic density and baryon asymmetry. We focus on the ratio λL/λXbecause we are interested in seeing if any relationship between these two theoretically unrelated quantities isrequired to obtain a particular relic abundance and asymmetry. In both cases shown, WIMPy leptogenesisgives the correct dark matter relic abundance and asymmetry when both couplings are O(1). Thus, a perfectlynatural choice of couplings, and the very same couplings that satisfy the WIMP miracle, can also generatethe correct baryon asymmetry if CP phases are large! Specifically, with mX = 4.25 TeV, mψ = 7.5 TeV,and |ǫ| = 0.075, the observational constraints are satisfied with λX = 2.7 and λL = 5.7; with mX = 1.5 TeV,mψ = 2.25 TeV, and |ǫ| = 0.0075, the couplings are λX = 2.8, λL = 2.5.

6Since the Standard Model phase transition is of second order, sphalerons do not suddenly shut off, and a more proper treatmentwould account for the gradual departure from equilibrium of the sphaleron effects. Since the asymmetry is typically generatedover a very short time period (we find numerically that it is on the order of ∆x ∼ 2 − 3 or ∆T ∼ 5 − 10 GeV), the dynamics ofsphaleron shut-off are largely irrelevant and the most important factor is the rate of L → B transfer at the washout freeze-outtime xwashout.

15

2YX = 3 x 10 -14

2YX =

9.4 x 10 -14HobservedL

2Yx =

3x

10 -13

1 2 3 4 5 61.0

10.0

5.0

2.0

3.0

1.5

7.0

ΛX

ΛLΛ

X

2YX =

8 x 10 -14

2YX=

2.5x

10 -13HobservedL

2Yx =

8x

10 -13

1 2 3 4 5 6

1.0

2.0

3.0

1.5

ΛX

ΛLΛ

X

Figure 8: Dark matter relic density (solid lines) and baryon asymmetry (dotted lines) as functions of couplingsλX and λL/λX . We consider two sets of massses, with mS = 5 TeV for both: (left) mX = 4.25 TeV, mψ = 7.5TeV, and |ǫ| = 0.075; (right) mX = 1.5 TeV, mψ = 2.25 TeV, and |ǫ| = 0.0075. The asymmetry contoursare, from top to bottom: (left) Y∆B = 5× 10−11, 8.5× 10−11 (observed asymmetry), and 3× 10−10; (right)Y∆B = 3× 10−11, 8.5× 10−11 (observed asymmetry), and 3× 10−10. The dark matter abundances are printedon the plots. In the shaded regions, the numerical value of ǫ is not consistent with our assumptions accordingto the bound (25).

In deriving our results, we assumed that dark matter only annihilates through lepton-number-violatinginteractions. In a more general model, dark matter may also have lepton-number-preserving interactions thatcontribute to the total annihilation cross section. We parameterize this possibility with the quantity

α ≡ 〈σXX→anything v〉〈σXX→Lψ v〉

≥ 1. (36)

When α > 1, the asymmetry generated by WIMPy leptogenesis is smaller, because only 1/α of dark matterannihilations proceed through lepton-number-violating couplings and can create an asymmetry7. As a result,the viable parameter space for WIMPy leptogenesis is reduced. In Figure 9, we show the masses mX , mψ

giving successful WIMPy leptogenesis with α > 1. In particular, we find that the lepton asymmetry fromWIMPy leptogenesis is too small in regions with large washout (mψ ∼ mS, where wash-out scattering is on-shell). While WIMPy leptogenesis is possible with lepton-preserving annihilation channels, mψ lies in a morerestricted region when α > 1.

To summarize, we have presented a model of WIMPy leptogenesis where the WIMP miracle has beenextended to the WIMPy baryogenesis miracle: the correct baryon asymmetry and WIMP relic density can begenerated simultaneously with TeV-scale masses and O(1) couplings. We find that, depending on the ratiomψ/mX and the Yukawa couplings, larger and smaller asymmetries are also possible over a range of aboutseven orders of magnitude. Generating the observed baryon asymmetry does require some correlation betweenmX and mψ, which may be explained if the masses have some common dynamical origin. For mX lighter thanabout 1 TeV, sphalerons decouple in the middle of asymmetry generation and the resulting baryon asymmetryis typically smaller than the observed Y∆B.

7When α > 1, the WIMP annihilation cross section is also larger, and λX , λL are smaller to give the same WIMP relic density.As discussed in Section 3.1.3, however, decreasing the couplings results in both a smaller washout rate and a smaller efficiency ofgenerating an asymmetry. These two effects counteract one another, and the change in couplings for α > 1 does not substantiallyaffect the asymmetry.

16

2mX < mΨ

Viable para

meters

Washout too strong

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

mX mS

mΨ

mS

mS = 5 TeV

2mX < mΨ

Viable para

meters

Washout too strong

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

mX mS

mΨ

mS

mS = 5 TeV

Figure 9: Regions in the mX -mψ plane for viable WIMPy leptogenesis with additional lepton-number-preserving dark matter annihilation modes: (left) α = 2, (right) α = 3. The masses giving the correctdark matter density and baryon asymmetry for some choice of perturbative coupling are shown in blue (mid-dle stripe). As the lepton-number-preserving annihilation cross section increases, the efficiency of asymmetrygeneration drops, and the marginal regions of parameter space become inaccessible, particularly the enhancedwashout region mψ ∼ mS . The descriptions of the regions on the plot are the same as those in Figure 7.

4 WIMP Annihilation to Quarks

4.1 Model Overview

If the final products of dark matter annihilation are quarks, WIMP annihilation can directly generate a baryonnumber asymmetry. The lower bound of ∼ TeV on mX in WIMPy leptogenesis (the dashed line in Figure 7)does not apply when WIMPs annihilate to quarks, since the production of baryon number no longer dependson efficient sphaleron interactions. Just as the leptogenesis model included new weakly charged vectorlikedoublets, this model requires new vectorlike colored states to couple to quarks. Such states can be pair-produced at the LHC, leading to much stronger constraints and better detection prospects, which we discussin Section 5.2.

The model content is similar to the leptogenesis model discussed in Section 3: vectorlike gauge singletdark matter X and X , singlet pseudoscalars Sα, and vectorlike exotic quark color triplets ψi and ψi. TheLagrangian is

L = Lkin + Lmass −i

2

(

λXαX2 + λ′XαX

2)

Sα + iλBα Sαuψ. (37)

A baryon asymmetry can be generated in u along with an equal negative baryon asymmetry in ψ. ψ mustdecay, because it would otherwise overclose the universe and violate bounds on stable colored particles.

The negative baryon asymmetry in ψ must not destroy the positive baryon asymmetry in u when it decays.This can happen in two ways:

1. ψ decays into a sector decoupled from Standard Model quarks at low energies. Total baryon number ispreserved, but the negative baryon number carried by ψ is sequestered from quarks at late times anddoes not eliminate the Standard Model asymmetry.

2. ψ decays into Standard Model quarks through baryon-number-violating couplings. The final baryonasymmetry is different from the asymmetry created initially in u from WIMP annihilations because ψdecays give an additional contribution to the baryon asymmetry.

We now implement each of the above scenarios.

1. ψi decays to light, baryon-number-carrying singlets ni plus Standard Model antiquarks. It can do sothrough a colored scalar φ with Standard Model gauge representation (3, 1,−1/3). The additional terms

17

X X ψ ψ S u d φ/d Q H n leptons

Z4 +i −i +1 +1 −1 −1 −1 −1 −1 +1 +1 +1

Table 2: Z4 charges of fields in models with WIMP annihilation to quarks.

in the Lagrangian are∆L = λi ψi di φ

∗ + λ′i φ di ni + h.c. (38)

A Z4 symmetry prevents ψ from decaying directly into Standard Model quarks through a QHψ termand eliminating the baryon asymmetry. We show the charges in Table 2. This Lagrangian has a U(1)3

flavor symmetry and satisfies all quark flavor constraints. In particular, ui, ψi, and ni have charges −1,+1, and −3, respectively, under the U(1)i factor of the flavor symmetry. φ has charge −2 under all U(1)flavor symmetries. We assume that the only sources of flavor violation are the Standard Model Yukawamatrices.

2. In this scenario, ψi also decays to two antiquarks plus a singlet n, but n is a Majorana fermion that doesnot carry any charge. Baryon number is now explicitly violated, and dark matter annihilations generate−1 unit of baryon number for each ψ + u produced from dark matter annihilations (because ψ → ddn).There is a new colored scalar di in the (3, 1,−1/3) representation of the Standard Model gauge groupthat mediates ψ decays. The additional terms in the Lagrangian are

∆L = λ ǫijk ψi dj d∗k + λ′i di di n+ h.c. (39)

A Z4 symmetry, with charges given in Table 2, prevents ψ from decaying to Standard Model quarksthrough other interactions that would destroy the baryon asymmetry. This Lagrangian can be naturallyrealized in supersymmetric models, where d is the down squark and n is the neutralino, although thisis not the only possible realization of this scenario. (39) has a U(3) flavor symmetry, which is thediagonal subgroup of the full U(3)u × U(3)d flavor group. The quark, ψi and di fields transform inthe fundamental of U(3). The Yukawa couplings between Suψ in (37) have a flavor-independent pieceand a flavor-dependent piece proportional to the up Yukawa matrix Yu, consistent with minimal flavorviolation.

In both scenarios, ψ decays to a singlet n plus quarks. Operators allowing ψ to decay entirely to quarks(such as φ∗d u or d∗d u) are forbidden by the Z4 symmetry. The Z4 symmetry also ensures the stability ofdark matter and of the proton. The proton is stable provided mp < 2mX ,mS because baryons have charge(−1)3 = −1 and can never decay into the lighter meson and lepton fields, which are uncharged under the Z4.

The Z4 symmetry in principle allows neutral baryons to oscillate into one another. For scenario #1, thegeneralized baryon number symmetry prohibits neutron-antineutron oscillation. In scenario #2, the baryon-number-violating term is antisymmetric in flavor indices, and the dominant contribution to neutron-antineutronmixing involves loops of W bosons and off-diagonal CKM matrix element insertions Vbd and Vsd. Since thebound on the neutron-antineutron oscillation operator c/Λ5(udd)2 is Λ & 10−100 TeV for c = 1 [15], the loop-and CKM-suppression is sufficient to lower the oscillation rate well below current constraints formS , mψ, md ∼TeV and O(1) couplings.

The Boltzmann equations for WIMP annihilation to quarks are changed only by group theory factors fromthe corresponding equations for leptons. Similarly, the chemical potentials relations are modified to reflect thenew interactions (38) or (39).

As with WIMPy leptogenesis, we assume that ψ is in equilibrium, and this places constraints on thecouplings through which it decays. In scenario #1, we considered the decay of ψ according to interactions givenin equation (38). If mψ > mφ, the ψ decay is two-body and the constraint (32) applies, giving λi & 6× 10−8.If mψ < mφ, ψ undergoes a three-body decay to d d n, and the constraint is

(λiλ′i)

2 &80

√g∗ T

2lepm

2φ

MPlm3ψ

. (40)

18

Washout

2mX < mΨ

Viable

paramete

rs

LHC gluino constraint

too strong

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

mX mS

mΨ

mS

mS = 1.5 TeV

2mX < mΨ

Viable

paramete

rs

Washout too strong

LHC gluino constraint0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

mX mS

mΨ

mS

mS = 5 TeV

Figure 10: Regions in the mX -mψ plane with the correct WIMP relic density and baryon asymmetry fromWIMPy baryogenesis, with (left) mS = 1.5 TeV and (right) mS = 5 TeV, and any choice of perturbativecouplings. The masses giving both observed abundances are shown in blue (middle stripe). The descriptionsof the regions on the plot are the same as those in Figure 7.

For Tlep = 50 GeV and mψ = 1 TeV, we find that the constraint on the geometric mean of the couplings is

√

λiλ′i & 6× 10−5

√

mφ

2 TeV. (41)

The constraints on scenario #2 are comparable.

4.2 Numerical Results

Because of its similarity to leptogenesis, we use the quark flavor structure of scenario #1 in our analysis, since itallows for a more direct comparison of numerical results in both cases. We consider two scenarios: mS = 5 TeV,to compare the results for quarks with that for leptons, and mS = 1.5 TeV, because dark matter can be muchlighter than in WIMPy leptogenesis since there are no constraints from sphaleron decoupling. For simplicity,we consider sphalerons to be out of equilibrium for the 1.5 TeV case and in equilibrium for the 5 TeV case toavoid considering sphaleron decoupling effects, although the calculation can be easily extended to include them.

Range of allowed masses: We show the range of allowed masses in Figure 10. Gluino searches at the LHCconstrain this scenario (see Section 5.2), in contrast with the leptogenesis model, for which the entire parameterspace is unconstrained by collider searches. This is particularly true for mS = 1.5 TeV, where LHC searcheswill cover almost the entire parameter space for dark matter annihilation to quarks during the 14 TeV run.The WIMP mass is already constrained to be mX & 295 GeV by the gluino bound discussed in Section 5.2along with the kinematic requirement that 2mX > mψ. This is true independent of all other parameters.

In both cases, the results are qualitatively similar to leptogenesis. With WIMP annihilation to quarks, theannihilation and washout cross sections are enhanced because the final states are charged under SU(3)C. Asa result, the baryon asymmetry is suppressed by the increased washout rate. This is partially offset by thefact that the self-energy contribution to ǫ is enhanced by a group theory factor as well. With mX = 0.9 TeV,the parameter space is actually larger than for mX = 4.25 TeV or WIMPy leptogenesis. Since in this case,sphalerons no longer inter-convert baryon and lepton number, the asymmetry created in quarks is distributedamong fewer fields, enhancing the asymmetry.

Range of allowed couplings: We do the same analysis as in Section 3.2. To compare the results of directbaryon asymmetry production with those for WIMPy leptogenesis, we choose a set of parameters used in theleptogenesis analysis: mX = 4.25 TeV, mψ = 7.25 TeV, mS = 5 TeV, and |ǫ| = 0.075. We also consider a

19

2YX = 3 x 10 -14

2YX =

9.5 x 10 -14HobservedL

2Yx =

3x

10 -13

1 2 3 4 51.0

10.0

5.0

2.0

3.0

1.5

7.0

ΛX

ΛBΛ

X 2YX =

10 -13

2YX =

4.4 x 10 -13HobservedL

2Yx =

10 -12

0.1 0.2 0.3 0.4 0.5

10

50

20

30

15

ΛX

ΛBΛ

X

Figure 11: Dark matter relic density (solid lines) and baryon asymmetry (dotted lines) as functions of couplingsλX and λB/λX . We consider two sets of massses: (left), mX = 4.25 TeV, mψ = 7.25 TeV, mS = 5 TeV,and |ǫ| = 0.075. The asymmetry contours are, from top to bottom: Y∆B = 4 × 10−11, 8.5× 10−11 (observedasymmetry), and 1.5 × 10−10. (right) mX = 0.9 TeV, mψ = 1.2 TeV, mS = 1.5 TeV, and |ǫ| = 0.075; theasymmetry contours are, from top to bottom: Y∆B = 5 × 10−11, 8.5 × 10−11 (observed asymmetry), and3× 10−10. In the shaded regions, ǫ is not consistent with our assumptions according to the bound (25).

corresponding point with light S and broken phase chemical potential relations: mX = 0.9 TeV,mψ = 1.2 TeV,mS = 1.5 TeV, and |ǫ| = 0.075. We show the results in Figure 11. The parameter points giving the correctdark matter density and baryon asymmetry are λX = 2.7, λB = 4.5 for mX = 4.25 TeV, and λX = 0.22,λB = 2.8 for mX = 0.9 TeV.

5 Experimental Constraints and Detection Prospects

In this section, we survey the possible experimental constraints and signals for models of WIMPy baryogenesis,considering both annihilation to leptons and annihilation to quarks. For WIMP annihilation to leptons, theexperimental bounds on mX and mψ are too weak to constrain leptogenesis because mX ,mψ & TeV arerequired to generate a sufficiently large baryon asymmetry. The prospects are better for WIMP annihilationto quarks, which predicts signals at indirect and direct detection experiments, as well as at the LHC. We firstgive a preview of our results in Table 3.

Annihilation to leptons Annihilation to quarks

Direct detection −− mX . 5 TeV for some parameters 8

(σX−nucleon ∼ 10−46 − 10−44 cm2)

Indirect detection mX . 200 GeV mX . 1 TeV

(antideuterons)

Colliders mψ . few hundred GeV, mψ . 1.44 TeV with

possible improvements 100 fb−1 LHC (14 TeV)

with targeted searches

EDM −− −−

Table 3: Search reach for minimal models of WIMPy baryogenesis/leptogenesis in current and near-futureexperiments. ‘−−’ indicates no signal in that search channel.

20

X X

q

X

q

Z/γ

Li

S

ψi

S

X X

e

S

e

X

ψ−

S

Figure 12: Feynman diagrams for dark matter scattering off (left) nucleons and (right) electrons in directdetection experiments for WIMPy leptogenesis.

X X

u†

S

u†

X

ψ

S

X X

nju†i

φ∗

dj

di

ψi

S

Figure 13: Feynman diagrams for dark matter scattering off nucleons in direct detection experiments whenWIMPs annihilate to quarks: (left) standard signal and (right) inelastic induced nucleon decay.

We provide details for each class of experiment in the following sections.

5.1 Dark Matter Detection

5.1.1 Direct Detection

Dark matter direct detection experiments are typically important probes of weak-scale dark matter models.As we show in this section, however, only WIMPy baryogenesis with dark matter annihilation to quarksis expected to give a signal in conventional direct detection experiments. This is because the dark matterscattering cross section is suppressed by loops of heavy fields, and it is only when dark matter couples directlyto quarks that the WIMP-nucleon cross section is large enough to give a signal in upcoming experiments. Weassume in this section that dark matter annihilates predominantly to first generation quarks/leptons. Thebaryon-number-violating interactions in WIMPy baryogenesis can also induce proton decay due to WIMPscattering, but we find that our models are consistent with all current and projected experimental bounds.

We first present the Feynman diagrams for the leading processes relevant to direct detection. With WIMPannihilation to leptons, we show the diagrams for scattering in direct detection experiments in Figure 12. Weshow the corresponding diagrams with WIMP annihilation to quarks in Figure 13.

WIMP annihilation to leptons: X can elastically scatter off electrons at one loop and nucleons at two loops.However, direct detection experiments are on the verge of testing dark matter models with nucleon scatteringat one loop [16, 2, 3, 5] and electron scattering at tree level [17]. Therefore, the elastic scattering signals fromour models are too small to be detected at near-future experiments.

As we discussed in Section 3.2, there can be lepton-number-preserving dark matter annihilation channels

8Precise reach depends on mψ , mS , λX , λB, and ǫ.

21

in addition to those responsible for baryogenesis, although the parameter space is more restricted in this case.The lepton-number-preserving WIMP interactions can lead to conventional WIMP signals in direct detectionexperiments, but do not probe the WIMP’s lepton-number-violating couplings, which are the crucial ingredi-ents of WIMPy leptogenesis.

WIMP annihilation to quarks: The dominant contribution to the direct detection cross section is the one-loopscattering of dark matter off the right-handed up quark. We estimate the dark matter-nucleon cross section:

σX−N ∼ 1

16π

(

λ2Bλ2X

16π2

)2µ2

m4X

, (42)

where µ is the reduced mass of the dark matter-nucleon system9. In our models, mX ≫ mN , where mN is thenucleon mass, so µ ≈ mN .

We determine the direct detection cross section for the benchmark points in Section 4.2. For the pointmX = 4.25 TeV,mψ = 7.25 TeV,mS = 5 TeV, λX = 2.7 and λB = 4.5, we find σX−N ≈ 1×10−44 cm2. For thepoint mX = 0.9 TeV, mψ = 1.2 TeV, mS = 1.5 TeV, λX = 0.22 and λB = 2.8, we find σX−N ≈ 4×10−46 cm2.The current limits from the XENON100, CDMS experiments [18, 19] on dark matter direct detection have aminimum bound of ∼ 10−44 cm2 for WIMPs with masses of ∼ 50 GeV. The upper limit on the cross sectionfor a TeV WIMP is ∼ 10−43 cm2. We therefore see that the cross sections for our benchmark points are belowcurrent bounds but are large enough that they can give a signal in upcoming direct detection experimentssuch as XENON1T [20]. We leave a detailed study of the direct detection reach for future work.

There also exists an inelastic scattering process that converts an up-type quark to two down-type anti-quarks, as we show in the right-hand graph in Figure 13. Such an inelastic process can lead to nucleon decaysinduced by WIMP scattering. The dominant process is X p→ X nπ+, along with the corresponding processeswith strange quark production (b quark production is kinematically suppressed). To avoid conflict with protondecay experiments, the induced proton decay rate should satisfy bounds outlined in [21]. Comparing our modelto the Hylogenesis model in [21], we find that the operator giving rise to induced proton decay in our model isdimension-9 (X2uddn/Λ5), whereas the corresponding Hylogenesis operator is dimension-7. At the hadroniclevel, the Hylogenesis process is 2 → 2, in contrast with our 2 → 3 process, which gives our model a relativephase space suppression ∼ 1/(2π)3. Furthermore, the Hylogenesis model has a dark matter mass ∼ O(GeV),while the dark matter mass in WIMPy baryogenesis is mX ∼ O(TeV). As a result, the dark matter numberdensity is smaller by a factor of (GeV/TeV) in WIMPy baryogenesis, and the incident flux of dark matterparticles is suppressed. Taking into account all factors, the induced proton decay rate for WIMPy baryogenesis

has a ∼ ( 12π )

3(

GeVTeV

)5 ∼ 10−17 suppression compared to that of Hylogenesis. Since induced proton decay ison the verge of current bounds for Hylogenesis models with a heavy scale Λ ∼ TeV, the proton lifetime inour model is safely above the current bound, and not within the reach of near-future proton decay experiments.

To summarize, models with WIMP annihilation to leptons typically predict the absence of a signal inconventional dark matter direct detection experiments, while models with WIMP annihilation to quarks haveWIMP-nucleon cross sections below the current bounds but accessible in upcoming experiments. Baryon/lepton-number-preserving WIMP interactions can also give a signal in direct detection experiments, but such modelshave a smaller viable parameter space. WIMP scattering can induce nucleon decay in WIMPy baryogenesismodels, but the proton decay rate is far lower than current experimental constraints.

5.1.2 Indirect Detection

Models of WIMPy baryogenesis have indirect detection prospects similar to those in conventional WIMPscenarios because the dark matter relic abundance is symmetric and is established by thermal freeze-out. Thisis in contrast with many asymmetric dark matter models, which typically have suppressed indirect detectionsignals due to the fact that dark matter is largely asymmetric today. The only asymmetric dark matter models

9Because the masses of all fields running in the loop are similar in mass, there is no significant mass suppression to the crosssection from evaluating the loop integral.

22

with indirect detection signals are those in which the symmetric component is regenerated after WIMP freeze-out [6]. In the following summary, we assume that WIMPs annihilate predominantly through the interactionsthat generate the baryon asymmetry.

We find that indirect detection is most promising with WIMPy baryogenesis with dark matter annihilatingto quarks. In this scenario, the final states are color-connected quarks and sterile fields ni, and the quarkshadronize in the dark matter rest frame. This populates the low-energy anti-deuteron spectrum, leading toa clean, low-background signal at GAPS and AMS-02. The mass reach in this scenario is mX . 1 TeV.Annihilation of dark matter in WIMPy leptogenesis also leads to qq production via Higgs decay, but thequarks hadronize in the Higgs rest frame. Fewer low-energy antideuterons are produced, and the mass reachis only mX . 200 GeV, which is too low for viable models of WIMPy leptogenesis. We give more details below.

WIMP annihilation to leptons: The indirect detection signals are energetic neutrinos, positrons, and secondary

photons from the leptons produced in WIMP annihilations, along with antiprotons and antideuterons (D) fromψ0 → h+ n→ bb+ n. Unfortunately, the dark matter mass of O(TeV) in leptogenesis gives a flux lower thanthe sensitivities of most upcoming indirect direction experiments. With the standard cross section for thermalWIMP annihilation (〈σann〉 ≈ 3× 10−26 cm3/s), the reach of most current experiments like Fermi-LAT [22] isin the mass range . O(100 GeV). One exception is in the scenario with a very steep dark matter profile inthe galactic center, which occurs in halo models favored by hydrodynamical simulations. In this case, HESSmeasurements of gamma rays from the galactic center are within a factor of two of constraining a 3 TeV WIMPwith standard annihilation cross section to leptonic final states [23]. Based on the HESS analysis, it is likelythat with more data Fermi-LAT could rule out WIMPy leptogenesis models with masses . few TeV if thesimilar assumptions on dark matter distribution are applied. Such constraints suffer from large uncertaintiesin the dark matter profile, however, and we caution that such strong limits on WIMP masses may not bepossible.

According to the general analysis performed in [24], the mass reach of low energy antideuteron detectionexperiments at AMS-02 and GAPS could be up to ∼ 1TeV if hadronization happens mostly in the rest frameof dark matter annihilation, as occurs in the gg channel in [24]. However, hadronic decay products in theleptogenesis scenario are secondary or tertiary, and hadronization typically happens in the boosted frame,similar to the WW channel in the same reference. The resulting mass reach could be only ∼ 200 GeV, whichis too low for WIMPy leptogenesis because sphalerons are decoupled during the era of asymmetry generationfor dark matter masses in this range.

WIMP annihilation to quarks: The possible signals are p, D, and γ. In contrast with leptogenesis, the baryonasymmetry can be generated after the electroweak phase transition and the dark matter mass can be as low as∼ 290 GeV according to the bound in Section 5.2. This is promising for detection at upcoming experiments,particularly low energy anti-deuteron searches. Because the primary products of WIMP annihilation nowinvolve color-connected u and ψ, a large proportion of hadronization proceeds in the rest frame, resulting ina larger rate of D production [24]. This extends the mass reach at GAPS and AMS-02 [25] to ∼ 1 TeV andcovers a large part of the WIMPy baryogenesis parameter space. Higher WIMP mass regions (∼ TeV) may beconstrained by Fermi-LAT gamma ray observations of the galactic center, but as discussed above with WIMPyleptogenesis, these constraints are highly dependent on the dark matter profile [23].

In general, models of WIMPy baryogenesis and leptogenesis satisfy all current constraints from indirectdetection experiments, and future searches for antideuterons are promising discovery channels for models withWIMP annihilation to quarks.

5.2 Collider Detection

We consider the LHC constraints and detection prospects for new charged particles predicted in WIMPybaryogenesis. We find that the LHC can strongly constrain the scenario with WIMP annihilation to quarksbut may not constrain WIMP annihilation to leptons. Searches for supersymmetry (SUSY) with missingenergy are relevant to our models, since WIMPy baryogenesis predicts new charged fields decaying to Standard

23

Figure 14: LHC electroweak pair production of ψ and its subsequent decay in the model with WIMP annihi-lation to leptons. ψ0 decays to a Higgs boson and the light neutral fermion nℓ, while ψ

± decays to W± andnℓ.

Model particles and neutral fermions. We focus on existing LHC searches for SUSY and leave for later workthe optimization of collider searches for the particular charged fields found in our models.

The strongest LHC constraints are bounds on new colored fields, such as gluinos and squarks. Currentsearches at the LHC therefore constrain the scenario with WIMP annihilation to quarks, which has new coloredfields ψ. The luminosity at the LHC is not yet large enough to bound electroweak production of new particles,due to the smaller production cross section, and softer jets and missing energy. As a result, current colliderconstraints on mψ in the scenario with dark matter annihilation to leptons are well below the range neededfor viable WIMPy leptogenesis. Higher luminosity and new targeted searches can improve the LHC reach formψ depending on its decay modes.

We now consider each scenario in more detail.

WIMP annihilation to leptons: A characteristic feature of the WIMPy leptogenesis model in section 3 isthe presence of an exotic vectorlike SU(2)L doublet ψ. The neutral and charged components of ψ can bepair-produced via electroweak gauge bosons. According to our arguments in Section 3.1.1, ψ decays promptly.

The dominant decay of ψ0 is to Higgs + nℓ through the interactions in (18), where nℓ is the light neutralmass eigenstate after Higgs-induced mixing between ψ and n. The resulting collider signature for pair pro-duction is ψ0ψ0 → 4b(4j) +ET. The charged component, ψ±, decays to W± + nℓ, with a collider signaturefor pair production of ψ+ψ− → W+W− +ET. The relevant diagrams are shown in Figure 14.

Searches at LEP constrain the masses of the charged and neutral components of ψ with bounds on pairproduction of charginos (χ± → W±χ0), mχ± & 100 GeV [26]. ψ± decays look identical to chargino decays,so mψ± & 100 GeV as well. Hadronic chargino decays, which have a 4j +ET final state, constrain the ψ0

mass. The LEP bound groups hadronic chargino decays with other decay modes, so the bound is not directlyapplicable to ψ0. A more careful analysis (that we leave for future work) is needed to determine the precisebound on ψ0, but we expect it to be on the order of 100 GeV as well. The bounds on both ψ± and ψ0 arewell below the typical mψ required for WIMPy leptogenesis.

With the current luminosity of 5 fb−1 at√s = 7 TeV, the LHC bounds the masses of weakly charged

particles appearing in cascade decays of colored particles, but does not constrain particles such as ψ that areonly produced directly from electroweak gauge bosons. Therefore, the LHC does not bound mψ at present,and the LEP constraint remains the most important. Searches for direct chargino and slepton production withfuture LHC data will improve the bounds on mψ to masses on the order of a few hundred GeV, but this isstill smaller than mψ needed in WIMPy leptogenesis.

New LHC searches at 14 TeV could possibly yield stronger constraints. For example, If the Higgs massis known, we could require a reconstruction of the Higgs mass among final state jet pairs, greatly reducingbackgrounds. If mψ ≫ mh, the final state Higgses are boosted and can be studied with jet substructuretechniques, as suggested in [27].

In summary, the collider constraints on ψ are currently too weak to place any bounds on WIMPy leptoge-nesis models. Future LHC running will improve the bounds on mψ, and we have outlined some of the possiblesignals here. A more detailed collider analysis is deferred to later work.

24

q

q

ψ

ψ†

φ∗

φ n†

d†

d†

n

d

d

g

Figure 15: LHC pair production of ψ and its subsequent decay in the model with WIMP annihilation toquarks.

WIMP annihilation to quarks: As with WIMP annihilation to leptons, there are new charged states at theweak scale. In this model, ψ carries color charge, and the bounds are consequently much stronger thanfor WIMPy leptogenesis: mψ & 590 GeV as inferred from the current LHC gluino search at 7 TeV. Thephenomenology depends on how ψ decays, and we outlined two possible models in (38) and (39). In both, ψdecays to two jets plus a singlet, and so the collider phenomenology is identical. For the purposes of notationin this section, we assume that ψ decays through an intermediate colored scalar φ to 2 Standard Model quarksand a singlet fermion, n.

ψ can be pair-produced at the LHC with the signature pp→ ψψ → 4j+ET. We show the relevant diagramin Figure 15.

Gluino searches at LHC7 bound the ψ mass. Both gluinos and ψ decay to jj +ET, and the bounds onboth gluinos and ψ are comparable, as their cross sections differ only by a group theory factor. We correctthe gluino bounds for this factor. We apply simplified model searches from ATLAS, which place bounds ongluino and squark masses in the presence of a massless neutralino [28]. The corresponding fields in WIMPybaryogenesis are ψ, the colored scalar φ, and the massless singlet fermion n. The lower bound on the ψ mass ismψ & 590 GeV, which comes from the gluino bound when the squark is much heavier than the gluino. In ourscenario, this means that mψ & 590 GeV when mφ ≫ mψ (numerically, mφ & 1.2 TeV). The bounds on mψ

cut significantly into the allowed parameter space for dark matter because 2mX > mψ, and so mX & 295 GeVfor heavy φ. The bounds are stronger for lighter φ because φ and ψ can be jointly produced. For example,the bound on mψ is about 30% higher for mφ = 1 TeV.

The LHC search reach for gluinos is expected to be mg ≈ 1.44 TeV at 100 fb−1 and√s = 14 TeV [29]

(with the assumptions of mSUGRA and heavy squarks), so models of WIMPy baryogenesis will be stronglyconstrained by future running of the LHC. The LHC will not reach the highest-mass regions of WIMPybaryogenesis, but will exclude models with masses mψ . 2 TeV, mX . 1 TeV, and O(1) couplings.

LHC searches also constrain the mass of the colored scalar φ. Since mφ is not directly relevant to theoutcome of WIMPy baryogenesis (apart from the requirement that it be light enough for ψ decays to bein thermal equilibrium), bounds on mφ do not directly constrain WIMPy baryogenesis. Nevertheless, theproduction rate of φ is comparable to that of squarks and is very high at the LHC. With the interaction(38), φ decays to di +ET and has an event topology identical to squark pair production in the MSSM: twojets (possibly b-tagged) plus missing energy. The current model-independent constraint is mφ & 875 GeV fordegenerate squarks of the first two generations [28]. In WIMPy baryogenesis, however, only a single field φ isnecessary, so the bound can be relaxed. Since φ can decay into b, the bound is approximately that of a sbottomsquark from DØ, mb > 250 GeV [30]. Future LHC running at 14 TeV will improve the bound to ∼ 2 TeV at

100 fb−1 [29], and has the potential to discover colored scalars in the mass range of WIMPy baryogenesis.

25

eL eR

eL

L†i

ψ†iSα Sβ

ψ†1

γ

eL eR

eL

Li

ψiSα Sβ

ψ†1

γ

eL eR

eL

L†i

ψ†iSβ Sα

ψ†1

γ

eL eR

eL

Li

ψiSβ Sα

ψ†1

γ

Figure 16: A set of two-loop contributions to the electron EDM that vanishes when summed together.

5.3 Electric Dipole Moment Constraints

A viable mechanism for baryogenesis necessitates the existence of new CP phases. Bounds on the electronand neutron electric dipole moments (EDMs) strongly constrain many new sources of CP -violation, but wefind CP phases in WIMPy baryogenesis are not constrained by EDM experiments. The minimal models ofWIMPy baryogenesis presented in this paper couple new fields to either left-handed or right-handed lightfermions (but not both), resulting in suppressed EDMs that are consistent with current observations. As aresult, minimal models of WIMPy baryogenesis do not have the CP problem often associated with models ofweak-scale physics.

In the models presented in Sections 3 and 4, the fields S and ψ couple exclusively to either left-handed orright-handed quarks and leptons. As a result, loops contributing to light fermion EDMs are helicity-preservingwith an even number of Yukawa couplings, half of which are of the form λα i and the other half λ∗β j . Bysumming over all permutations of different flavors of S, L and ψ on the internal lines, it can be shown that allone- and two-loop diagrams appear in pairs that are complex conjugates of one another. Summing over eachset of pairs leads to a result that is real, and hence a vanishing EDM. As an example, we show in Figure 16 aset of four diagrams contributing to the electron EDM: the sum of the first two is proportional to λLα1λ

∗Lβ1,

while the sum of the second two is proportional to λ∗Lα1λLβ1. Therefore, the sum of all four is real and doesnot contribute to the electron EDM.

The two-loop EDM in the Standard Model vanishes for the same reason as in WIMPy baryogenesis:CP -violation arises only in couplings to one chirality of fermion. In both the Standard Model and WIMPybaryogenesis, the neutron EDM is non-zero at three loops, and we show the relevant diagrams in Figure 17.The principal difference between the two is that CP violation vanishes in the Standard Model with fewer thanthree generations, so the Standard Model EDM is suppressed by mixings involving all three generations. Bycontrast, WIMPy baryogenesis has a contribution to the EDM with only two generations of quarks that coupleto more than one flavor of S, and if the model is minimally flavor violating, the EDM will be suppressedby sin2 θc ≈ 0.05, the square of the Cabibbo angle. The naıve estimate for the neutron EDM in WIMPybaryogenesis with O(1) couplings is

dne

∼ sin2 θc(16π2)3

mu

m2S

. (43)

Substituting mS ∼ 5 TeV and mf ∼ MeV gives dn/e . 5 × 10−32 cm, which is well below the currentexperimental bound of dn/e < 2.9 × 10−26 cm [31]. The electron EDM from WIMPy leptogenesis is evensmaller than this, as flavor-changing effects in the charged lepton sector are suppressed by neutrino masses,and the EDM is also well below the experimental bound of de/e < 1.05× 10−27 cm [32].

Phases fromWIMPy leptogenesis can also contribute to EDMs via other new, weak-scale fields not includedin our minimal models. Since these contributions are model-dependent, we do not consider them further.

Minimal models of WIMPy baryogenesis do not suffer from a CP problem and are consistent with low-energy experiments, but it is possible that other, model-dependent contributions to the EDM could be con-strained by and give rise to signals in electron and neutron EDM experiments.

26

d

W W

c t

g

bd

u

S1 S2

ψ†u

ψ†c

g

cu

Figure 17: (left) Three loop EDM in the Standard Model. (right) The analogous diagram for quark EDMsin WIMPy baryogenesis. The photon line attaches to any charged internal fields.

6 Conclusions

In this paper, we explored a novel scenario called WIMPy baryogenesis that extends the WIMP miracle bygenerating the observed baryon asymmetry through annihilations of weak-scale dark matter and provides adynamical connection between the dark matter and baryon abundances. We found that natural couplings andweak-scale masses for the new fields can lead to the correct baryon asymmetry. As a by-product of linkingbaryogenesis with dark matter annihilation, we introduced a new mechanism for weak-scale baryogenesis,avoiding any conflicts with reheat bounds in supersymmetric theories.

The key observation is that, if dark matter annihilation proceeds via CP− and (Standard Model) B− orL−violating operators, all Sakharov conditions for baryogenesis are satisfied. Successful models also suppresswashout prior to dark matter freeze-out. In our models, such suppression results from the heaviness of thefield ψ carrying Standard Model gauge charges and B− or L−number that is one of the final states in darkmatter annihilation. Additional discrete symmetries forbid such exotic fields from decaying back to StandardModel fields. We presented models where dark matter annihilates to either quarks or leptons, and found viableparameter spaces with natural couplings and TeV-scale masses in both scenarios.

In models where dark matter annihilates to leptons, the lepton asymmetry must be generated before theelectroweak phase transition so that the asymmetry can be transferred to baryons via sphalerons. As a result,dark matter and ψ masses must be O(TeV). Because the new states are heavy and dark matter does notcouple directly to quarks, dark matter in this set-up is not in reach of near-future direct and indirect detectionexperiments. In this scenario, ψ is charged only under weak interactions, making LHC searches challenging,although targeted searches at high integrated luminosity may allow discovery.

If dark matter annihilates to quarks, baryogenesis can occur after the electroweak phase transition, allowingsmaller dark matter and ψ masses. With lighter new states and colored objects, dark matter in these modelscan be within reach of direct detection experiments and indirect detection searches for antideuterons. LHCsearches for ψ are similar to gluino searches and can exclude mψ . 1.44 TeV at 100 fb−1 and

√s = 14 TeV.

In both scenarios, WIMPy baryogenesis models can generate both the correct dark matter relic densityand the baryon asymmetry at the weak scale. Such models predict new weak-scale particles that can lead tosignals in dark matter direct and indirect detection experiments, and that may be accessible at the LHC.

Acknowledgements

We wish to thank Zackaria Chacko, Tongyan Lin, Raman Sundrum, David Simmons-Duffin, and Neal Weinerfor helpful conversations. LR would like to thank the Aspen Center for Physics and the KITP for theirhospitality during progress on this work. Some of the numerical calculations in this paper were performed onthe Odyssey cluster supported by the FAS Research Group at Harvard University. Feynman diagrams weredrawn using JaxoDraw [33]. This work is supported by NSF grant PHY-0855591 and the Harvard Center forthe Fundamental Laws of Nature. YC is also supported in part by NSF grant PHY-0801323 and the MarylandCenter for Fundamental Physics.

27

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